A Combinatorial Approach to Nonlocality and Contextuality

Abstract

Abstract. So far, most of the literature on (quantum) contextuality and the Kochen-Specker theorem seems either to concern particular examples of contextuality, or be considered as quantum logic. Here, we develop a general formalism for contextuality scenarios based on the combinatorics of hypergraphs which significantly refines a similar recent approach by Cabello, Severini and Winter (CSW). In contrast to CSW, we explicitly include the normalization of probabilities, which gives us a much finer control over the various sets of probabilistic models like classical, quantum and generalized probabilistic. In particular, our framework specializes to (quantum) nonlocality in the case of Bell scenarios, which arise very naturally from a certain product of contextuality scenarios due to Foulis and Randall. In the spirit of CSW, we find close relationships to several graph invariants. The recently proposed Local Orthogonality principle turns out to be a special case of a general principle for contextuality scenarios related to the Shannon capacity of graphs. Our results imply that it is strictly dominated by a low level of the Navascués-Pironio-Acín hierarchy of semidefinite programs, which we also apply to contextuality scenarios.We derive a wealth of results in our framework, many of these relating to quantum and supraquantum contextuality and nonlocality, and state numerous open problems. For example, we show that the set of quantum models on a contextuality scenario can in general not be characterized in terms of a graph invariant.In terms of graph theory, our main result is this: there exist two graphs G 1 and G 2

Generalized Satisfiability Problems via Operator Assignments

“…Second, we apply pp-definability to the problem of quantum realizability of contextuality scenarios. Recently, Fritz [12] used Slofstra's results [24] to resolve two problems raised by Acin et al in [1]. Using pp-definability and Slofstra's results, we obtain new proofs of Fritz's results that have the additional feature that the parameters involved are optimal.…”

“…Note that P 1-IN-3 SAT (X 1 , X 2 , X 3 ) = 3 4 X 1 X 2 X 3 + 1 4 X 1 X 2 + 1 4 X 2 X 3 + 1 4 X 1 X 3 − 1 4 X 1 − 1 4 X 2 − 1 4 X 3 + 1 4 , so the difference is that, even though the characteristic polynomial equation P 1-IN-3 SAT (X 1 , X 2 , X 3 ) = −I is satisfied by an operator assignment if and only if the resolution of the identity equation 1 2…”

“…is satisfied by the same operator assignment, the two polynomials P 1-IN-3 SAT (X 1 , X 2 , X 3 ) and 1 2 (1−X 1 )+ 1 2 (1−X 2 )+ 1 2 (1−X 3 ) are by no means the same.…”

Zoology of Atlas-Groups: Dessins D’enfants, Finite Geometries and Quantum Commutation

“…The near hexagon O − 8 (2) on 765 points is described in the text. (12,92,12,3) [120 28 , 1120 3 ] (3) , srg, NO + (8, 2), pg (7,8,4) 0.817 (15,99,11,6) [135 64 , 960 9 ] (3) , srg, G 4 , pg (8,7,4), 3QB * 0.770 (96,624,72,85) [960 36 , 4320 8 ] (4) 0.923 (4,6). Since the permutation representation is a subgroup of the modular group Γ = PSL(2, Z), it is possible to see the dessin D as an hyperbolic polygon D H .…”

“…Over the last few years, it has been recognized that the detailed investigation of commutation between the elements of generalized Pauli groups-the qudits and arbitrary collections of them [1]-is useful for a better understanding of the concepts of quantum information, such as error correction [2,3], entanglement [4,5] and contextuality [6][7][8], that are cornerstones of quantum algorithms and quantum computation. Only recently, the first author observed that much of the information needed is encapsulated in permutation representations, of rank larger than two, available in the Atlas of finite group representations [9].…”